Finite Group Databasebeta

← #30637673

Subgroups — #30637673

Order 16256.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550463C211yesmaximal, characteristic, complemented
2#22550027C211yesmaximal, characteristic, complemented
3#22549784C211yesmaximal, characteristic, complemented
4#22549614C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#447×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C2×#14523236—21no—
10#16861204—21no—
11#16861195—21no—
12#16861193—21no—
13#16861188C2211yescharacteristic
14#16861181—21no—
15#16861177—21no—
16D4×Dic127—21no—
17C2×C22⋊Dic127—21no—
18C2×(C2×D127)⋊C4C2211yescharacteristic
19#16861130—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C127⋊#95C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×D8×C127—21no—
24#121×C127C2211yescharacteristic
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254D411yescharacteristic, complemented
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—42no—
32C2×C22⋊D127—42no—
33C2×C22⋊D127—42no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37#14523236—42no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127D411yescharacteristic, complemented
41Dic127⋊C4—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—42no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49C4×Dic127—21no—
50C24×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—42no—
55C2×D4×C127D411yescharacteristic, complemented
56C22×C4×C127C2311yescharacteristic
57D8×C127—42no—
58C2×C8×C127—21no—
59C4⋊C4×C127D411yescharacteristic, complemented
60C22⋊C4×C127—21no—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—41no—
65C22×D127—41no—
66Dic254—41no—
67C22⋊D127—48no—
68C22⋊D127—81no—
69C22⋊D127—81no—
70C2×Dic127—22no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—81no—
78C127⋊C8—41no—
79C23×C127—22no—
80C23×C127—21no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127C2×D411yescharacteristic
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—41no—
91D4×C127—41no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95C2×C4×C127—21no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C8×C127—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—42no—
107C22×C127—41no—
108C22×C127—81no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127—81no—
112C22×C127#12011yescharacteristic
113C22×C127#13511yescharacteristic
114C22×C127AGL(1,ℤ/8)11yescharacteristic
115C2×D127—42no—
116C2×D127—41no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—81no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—41no—
123Dic127—42no—
124Dic127—41no—
125Dic127—41no—
126Dic127—41no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—81no—
130C2×C127—82no—
131C2×C127#44911yescharacteristic
132C2×C127#45311yescharacteristic
133C2×C127B(4,𝔽2)11yescharacteristic
134D127—81no—
135#1600—1271nomaximal, Sylow
136C127#160011yescharacteristic, complemented, Sylow
137#538—1271no—
138#450—1271no—
139#447D12711yescharacteristic, complemented
140#392—1271no—
141#328—1271no—
142#327—1271no—
143#323—1271no—
144C22×D4—2541no—
145C22×D4C2×D12711yescharacteristic
146C2×SD16—2541no—
147C2×D8—21no—
148#126—2541no—
149#124—2541no—
150#123—2541no—
151C22⋊Q8—1271no—
152#121—2541no—
153#121C2×D12711yescharacteristic
154#120—2541no—
155C4×D4—2541no—
156C2×C22⋊C4—1271no—
157C2×C22⋊C4—2541no—
158#102—21no—
159#102—2541no—
160C22⋊C8—1271no—
161C22⋊C8C2×D12711yescharacteristic
162#95—1271no—
163C24—21no—
164C24—1271no—
165C2×Q8—1271no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—5082no—
171C2×D4—5082no—
172C2×D4—5082no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4C22⋊D12711yescharacteristic, complemented
176C22×C4—1271no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180SD16—5082no—
181D8—42no—
182C2×C8—21no—
183C2×C8—2541no—
184C4⋊C4—1271no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4C22⋊D12711yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—5082no—
190C22⋊C4—2541no—
191C22⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C42—2541no—
197C23—22no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—5081no—
206C23—5081no—
207C23—2541no—
208C23—2541no—
209C23—2541no—
210C23D4×D12711yescharacteristic
211Q8—5081no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—10161no—
220D4—10161no—
221D4—5088no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—10161no—
225C2×C4—5081no—
226C2×C4—5081no—
227C2×C4—5081no—
228C2×C4—2542no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—10161no—
249C22—5082no—
250C22—5081no—
251C22—5081no—
252C22—5081no—
253C22#1686117711yescharacteristic
254C22#1686122811yescharacteristic
255C22#1686123311yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—5082no—
260C4—5081no—
261C4—5081no—
262C4—5081no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2—81no—
267C2—10161no—
268C2#2255001811yescharacteristic
269C2#2255004211yescharacteristic
270C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#447×C127C2
socleC22×C127#120
hypercenter#447D127
hypocenterC127#1600
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550463C2semidirect
2. #22550027C2semidirect
3. #22549784C2semidirect
4. #22549614C2semidirect
5. #22549610C2semidirect
6. #22549599C2semidirect
7. #447×C127C2semidirect
28. C2×Dic254D4semidirect
40. C4⋊Dic127D4semidirect
55. C2×D4×C127D4semidirect
59. C4⋊C4×C127D4semidirect
136. C127#1600semidirect
139. #447D127semidirect
175. C2×D4C22⋊D127semidirect
187. C4⋊C4C22⋊D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225504632
#225500272
#225497842
#225496142
#225496102
#225495992
#447×C1272
#1600127